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Reciprocal matrices: properties and approximation by a transitive matrix

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Abstract

Reciprocal matrices and, in particular, transitive matrices, appear in several applied areas. Among other applications, they have an important role in decision theory in the context of the analytical hierarchical process, introduced by Saaty. In this paper, we study the possible ranks of a reciprocal matrix and give a procedure to construct a reciprocal matrix with the rank and the off-diagonal entries of an arbitrary row (column) prescribed. We apply some techniques from graph theory to the study of transitive matrices, namely to determine the maximum number of equal entries, and distinct from \(\pm 1\), in a transitive matrix. We then focus on the n-by-n reciprocal matrix, denoted by C(n, x),  with all entries above the main diagonal equal to \(x>0.\) We show that there is a Toeplitz transitive matrix and a transitive matrix preserving the maximum possible number of entries of C(n, x),  whose distances to C(n, x),  measured in the Frobenius norm, are smaller than the one of the transitive matrix proposed by Saaty, constructed from the right Perron eigenvector of C(n, x). We illustrate our results with some numerical examples.

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Authors and Affiliations

  1. CMUC and Departamento de Matemática, Universidade de Coimbra, 3001-454, Coimbra, Portugal

    Natália Bebiano

  2. CMA and Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516, Caparica, Lisbon, Portugal

    Rosário Fernandes

  3. CEAFEL and Faculdade de Economia do Porto, Universidade do Porto, 4200-464, Porto, Portugal

    Susana Furtado

Authors
  1. Natália Bebiano
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  2. Rosário Fernandes
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  3. Susana Furtado
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Corresponding author

Correspondence to Susana Furtado.

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Communicated by Jinyun Yuan.

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N. Bebiano: This work was partially supported by project UID/MAT/00324/2019. R. Fernandes: This work was partially supported by project UID/MAT/00297/2019. S. Furtado: This work was partially supported by project UID/MAT/04721/2019.

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Bebiano, N., Fernandes, R. & Furtado, S. Reciprocal matrices: properties and approximation by a transitive matrix. Comp. Appl. Math. 39, 50 (2020). https://doi.org/10.1007/s40314-020-1075-2

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